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Theorem ovresd 7600
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
ovresd.1 (𝜑𝐴𝑋)
ovresd.2 (𝜑𝐵𝑋)
Assertion
Ref Expression
ovresd (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Proof of Theorem ovresd
StepHypRef Expression
1 ovresd.1 . 2 (𝜑𝐴𝑋)
2 ovresd.2 . 2 (𝜑𝐵𝑋)
3 ovres 7599 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
41, 2, 3syl2anc 584 1 (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108   × cxp 5683  cres 5687  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-res 5697  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  sscres  17867  fullsubc  17895  fullresc  17896  funcres2c  17948  rngchom  20623  ringchom  20652  rhmsubclem4  20688  irinitoringc  21490  psmetres2  24324  xmetres2  24371  prdsdsf  24377  xpsdsval  24391  xmssym  24475  xmstri2  24476  mstri2  24477  xmstri  24478  mstri  24479  xmstri3  24480  mstri3  24481  msrtri  24482  tmsxpsval  24551  ngptgp  24649  nlmvscn  24708  nrginvrcn  24713  nghmcn  24766  cnmpt1ds  24864  cnmpt2ds  24865  ipcn  25280  caussi  25331  causs  25332  minveclem2  25460  minveclem3b  25462  minveclem3  25463  minveclem4  25466  minveclem6  25468  ftc1lem6  26082  ulmdvlem1  26443  abelth  26485  cxpcn3  26791  rlimcnp  27008  hhssnv  31283  madjusmdetlem3  33828  qqhcn  33992  qqhucn  33993  ftc1cnnc  37699  ismtyres  37815  isdrngo2  37965  naddcnffo  43377
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